#7664891

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3475	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(15,21)(16,23)(18,25)(19,24);(1,2)(3,4)(5,7,9,10)(6,8,11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28);(5,8)(6,10)(7,11)(9,12)(13,14,17,20)(15,18,21,25)(16,19,23,24)(22,26,27,28);(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,27)(24,28);(5,6)(7,8)(9,11)(10,12)(13,14)(15,18)(16,24)(17,20)(19,23)(21,25)(22,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,22)(17,23)(18,26)(20,24)(21,27)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572087);